Hi together,

I was part of the live stream yesterday where you asked for some whiteboard results why a 10 minute launch delay would be bad - so I tried to bring it down to some mathematical arguments.

First of all, please excuse any spelling mistakes and also take a note that I'm nowhere near a rocket scientist but rather someone who likes KSP and is interessted in orbital mechanics and some of the values are simplified / guesswork (I tried to mark guesswork with

).

So to answer the question why a 10 minute launch delay is bad we need to take a look at some facts at first. To do any change in the orbit a space craft needs what's called delta-v, the potential to change in velocity. Given the mission profile flown with CrewDragon demo-1 we just need to care about the capsules delta-v budget - so let's start by looking into what we have.

By looking into

https://www.faa.gov/about/office_org/headquarters_offices/ast/environmental/nepa_docs/review/launch/media/fonsi_dragon_pad_abort.pdf we can find, that Crew Dragon capsule carries ~1388kg (3060 pounds) of NTO/MMH fuel. Given it's dry mass of 9525kg from wikipedia we can use the ideal rocket equation to calculate the total available delta-v for it.

dv = v

_{e} * ln ( m

_{0} / m

_{1} )

where

m

_{0} is the initial mass of 9525kg + 1388kg of fuel = 10913kg

m

_{1} is the final dry mass of 9525kg

v

_{e} is the exhaust velocity in vacuum of NTO/MMH of 3347m/s

so dv = 3347 m/s * ln(10913 / 9525) =

455 m/sFrom the Demo-1 launch the capsule is brought to a roughly 280km by 280km

orbit by the second stage and is then finally separated from the Falcon-9 this is why I only look into delta-v budget of the Dragon capsule. The 280km temporary orbit is something I'm unsure of but I found resources in form of TLEs (

https://sattrackcam.blogspot.com/2020/05/the-trajectory-of-upcoming-crew-dragon.html) which lead to a rough 280km orbit and looking into Space-X mission profile on

https://www.spacex.com/launches/?utm_source=morning_brew it's seems that they are doing a different phasing which might cost some more energy as my calculated Hohmann transfer down below.

The ISS orbits earth in a 400 by 420km orbit, for simplifying I'll take a 400km circular orbit as those 20km just add around 12m/s for the transfer.

So, completly independent of being 10 minutes late or not CrewDragon needs to perform a transfer from its 280km orbit to the ISS orbit. I calculated the energy needed for this using the calcuations I found here:

https://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/cami/library/online_libraries/aerospace_medicine/tutorial/media/III.4.1.5_Maneuvering_in_Space.pdfRemoving the 68m/s of delta-v from our 455m/s and we are left with

387m/s for flying around and deorbiting.

Speaking of deorbiting, the capsule must come back at some point in time which will use up some fuel. Some users at stack exchange did some math for the space shuttle (

https://space.stackexchange.com/questions/12011/how-could-a-90-m-s-delta-v-be-enough-to-commit-the-space-shuttle-to-landing) which it turns out that around 90m/s can be enough to get back, so lets remove that from our capsule as well and we are left with 297m/s for orbital shenanigans.

Up until this point it's all the same for a correctly timed launch or a 10 minute late launch. After 10 minutes the ISS with it's orbital velocity of around 7,673km/s will have travelled 4600km from it's optimal point but as the capsule is lower it will slowly catch up - so it's just a bit more of waiting for the astronauts to get to the ISS - no big deal as Wikipedia states the CrewDragon is laid out for a week in space or 210 days docked to the ISS.

But - while waiting for 10 minutes on the launch pad - not only does the ISS travel 4600km but the earth also rotates away from the ISS.

Given that the earth rotates by 360°/day it turns by 0,25° per minute (2,5° in 10 minutes) so the orbit will be misaligned by 2,5° waiting for 10 minutes.

More specifically for a launch to the same inclination the right ascension of the ascending node will differ by 2,5° and if launched pointing towards the ISS their inclination will differ by those 2,5°.

The RAAN (right ascension of the ascending node) is the angle in the equator between the ascending node (where the orbit crosses the equator from south to north) and the point of vernal equinox.

To cope with those misalignments CrewDragon needs to perform a so called plane change which covers either the inclination change and the change of the RAAN depending on where you do the burn.

Given a simple plane change where only the direction changes the equation is rather simple the forumla looks like this:

dV = 2 * dV

_{initial} * sin( theta / 2 )

where

dV

_{initial} is the current orbital speed and

theta is the plane change angle

As this burn is normally performed prior to the transfer burn let's calculate it for the 280km orbit:

dV = 2 * 7,742 km/s * sin (2,5°/2) = 0,337km / s =

337 m/s And our 297m/s left over are busted - and those were only there because we calculated for a completly empty capsule.

The problem with plane changes is that it's not just speeding up but you are going really fast in one direction (7,742 km/s) and you have to stop and accelerate in a different direction again.

(This is also covered by the PDF i linked in above)

Given this ideal and simplified conditions delaying the launch by up to 8,8 minutes works and after that the capsule does not have enough fuel to reach to ISS by any means.

Having this really costly plane change manuever is also the reason why they need to wait up until Saturday now because they can only launch when the ISS passes over the launch site directly.

Hope this shed some light and I also hope I did not made any big errors. (If you do the calcuations for a lower 200 by 200km orbit the transfer takes 110m/s and the plane change 339m/s also busting the budget so the initial orbit should not make a big difference)